1# Graphing techniques 2 3## Reciprocal continuous functions 4 5If $y=f(x)$, the reciprocal function is: 6 7$$y={1 \over f(x)}$$ 8 9As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (vert asymptote at $f(x)=0$) 10 11<!-- As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$ --> 12 13- reciprocal functions are always on the same side of $x=0$ 14- if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$ 15- point of inflection at $P(1,1)$ 16 17## Locus of points 18 19- set of points that satisfy a given condition 20- path traced by a point that moves according to a condition 21- graph on CAS - **conics** 22 23### Circular loci 24 25point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre) 26 27 28$$PC = r$$ 29 30$$(x-a)^2 + (y-b)^2 = r^2$$ 31 32 33 34### Linear loci 35 36$$QP = RP $$ 37$$\sqrt{(x_Q-q_P)^2+(y_Q-y_P)^2} = \sqrt{(x_R-x_P)^2+(y_R-y_P)^2}$$ 38 39points $Q$ and $R$ are fixed and have a perpendicular bisector $QR$. Therefore, any point on line $y=mx+c$ is equidistant from $QP$ and $RP$. 40 41Since the bisector of the line joining points $Q$ and $R$ is perpendicular to $QR $: 42 43$$m( QR ) \times m( RP ) = -1$$ 44 45### Parabolic loci 46 47$$PD = PF $$ 48$$|y-z|=\sqrt{(x-x_F)^2+(y-y_F)^2}$$ 49$$(y-z)^2=(x-x_F)^2+(y-y_F)^2$$ 50 51Distance of point $P(x,y)$ from fixed point $F(a,b)$ is equal to the distance of $P$ from $y=z \perp$. 52 53Fixed point $F$ is the **focus** (halfway between $y=z$ and $y=y_P$) 54 55Fixed line $x=z$ is the **directrix** 56 57### Elliptical loci 58 59Point $P$ moves so that the sum of its distances from two fixed points $F_1$ and $F_2$ is a constant $k$. 60 61$${F_1 P} + F_2 P =k$$ 62 63**Two** foci at $F_1$ and $F_2$ 64 65Cartesian equation for ellipses: 66$${(x-h)^2 \over a^2} + {(y-k)^2 \over b^2} = 1$$ 67centered at $(h,k)$. Width is $2a$, height is $2b$. 68 69### Transformations 70$$(x,y) \rightarrow (x \prime, y \prime)$$ 71 72where $x \prime$ and $y \prime$ are the transformation factors (dilation away from $x$-axis means coefficient of $y$ increases in $y \prime$, and vice versa). 73 74Transformed equation is the same as initial equation with each term divided by its dilation coefficients (must be in terms of $x\prime$ and $y\prime$). 75 76e.g. 77 78$x^2 + y^2 = 1$ is dilated $3$ from $x$, $5$ from $y$. 79Transformation rule is $(x\prime,y\prime) = (5x,3y)$ 80$x={x\prime \over 5},\quad y={y\prime \over 3}$ 81 82Equation $x^2 + y^2=1$ becomes 83 84$${(x\prime)^2 \over 25}+ {(y\prime)^2 \over 9}=1$$ 85 86 87### Hyperbolic loci 88 89$$|(F_2P - F_1P )| = k$$ 90 91Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors): 92$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$ 93 94Asymptotes at $y=\pm {b \over a}(x-h)+k$ 95To make hyperbola up/down rather than left/right, swap $x$ and $y$ 96 97$y^2-x^2=1$ produces hyperbola shifted 90 $^\circ$ (top and bottom of asymptotes) 98 99## Parametric equations 100 101Parametric curve: 102 103$$x=f(t), \quad y=g(t)$$ 104 105$t$ is the parameter 106 107## Polar coordinates 108 109$$x = r\cos\theta, \quad y = r\sin\theta$$ 110 111### Spirals 112$$r={\theta \over n\pi}$$ 113- solve intercepts for multiples of $\pi \over 2$ 114- or draw table of values for $r$ and $\theta$ for each $n\pi \over 2$ 115 116### Circles 117$$r=a$$ 118 119### Lines 120 121Horizontal: $r={n \over \sin \theta}$ 122Vertical: $r={n \over \cos \theta}$ 123 124### Solving polar graphs 125 126solve in terms of $r$ 127 128e.g. $x=4$ 129$r\cos\theta = 4$ 130$r={4 \over \cos\theta}$ 131 132e.g. $y=x^2$ 133$r\sin\theta = r^2 \cos^2\theta$ 134$\sin \theta = r \cos^2\theta$ 135$r = {\sin \theta \over \cos^2\theta} = \tan\theta \sec\theta$ 136 137e.g. $r=6\cos \theta\quad$ *(multiple by $r$)* 138$r^2=6r\cos\theta$ 139$x^2+y^2=6x$ 140complete the square 141 142## Other graphs 143 144### Cardioids 145 146$$ 147 148### Roses